<< Chapter < Page Chapter >> Page >

3 x 2 75

3 ( x 5 ) ( x 5 )

Got questions? Get instant answers now!

a 3 b 4 m a m 3 n 2

a m ( a b 2 + m n ) ( a b 2 m n )

Got questions? Get instant answers now!

Fundamental rules of factoring

There are two fundamental rules that we follow when factoring:

    Fundamental rules of factoring

  1. Factor out all common monomials first.
  2. Factor completely.

Sample set b

Factor each binomial completely.

4 a 8 b - 36 b 5 . Factor out the common factor 4b. 4 b ( a 8 - 9 b 4 )

Now we can see a difference of two squares, whereas in the original polynomial we could not. We’ll complete our factorization by factoring the difference of two squares.

4 a 8 b - 36 b 5 = 4 b ( a 8 - 9 b 4 ) = 4 b ( a 4 + 3 b 2 ) ( a 4 - 3 b 2 )

Got questions? Get instant answers now!

x 16 - y 8 . Factor this difference of two squares . x 16 - y 8 = ( x 8 + y 4 ) Sum of two squares Does not factor ( x 8 - y 4 )  Difference of two squares Factor it! = ( x 8 + y 4 ) ( x 4 + y 2 ) ( x 4 - y 2 ) Factor again! = ( x 8 + y 4 ) ( x 4 + y 2 ) ( x 2 + y ) ( x 2 - y )

Finally, the factorization is complete.

These types of products appear from time to time, so be aware that you may have to factor more than once.

Got questions? Get instant answers now!

Practice set b

Factor each binomial completely.

m 4 n 4

( m 2 + n 2 ) ( m n ) ( m + n )

Got questions? Get instant answers now!

16 y 8 1

( 4 y 4 + 1 ) ( 2 y 2 + 1 ) ( 2 y 2 1 )

Got questions? Get instant answers now!

Perfect square trinomials

Recall the process of squaring a binomial.

( a b ) 2 a 2 2 a b b 2 ( a b ) 2 a 2 2 a b b 2

Our Method Is We Notice
Square the first term. The first term of the product should be a perfect square.
Take the product of the two terms and double it. The middle term of the product should be divisible by 2 (since it’s multiplied by 2).
Square the last term. The last term of the product should be a perfect square.

Perfect square trinomials always factor as the square of a binomial.

To recognize a perfect square trinomial, look for the following features:

  1. The first and last terms are perfect squares.
  2. The middle term is divisible by 2, and if we divide the middle term in half (the opposite of doubling it), we will get the product of the terms that when squared produce the first and last terms.

In other words, factoring a perfect square trinomial amounts to finding the terms that, when squared, produce the first and last terms of the trinomial, and substituting into one of the formula

a 2 + 2 a b + b 2 = ( a + b ) 2 a 2 - 2 a b + b 2 = ( a - b ) 2

Sample set c

Factor each perfect square trinomial.

x 2 + 6 x + 9 . This expression is a perfect square trinomial. The x 2 and 9 are perfect squares.

The terms that when squared produce x 2 and 9 are x and 3, respectively.

The middle term is divisible by 2, and 6 x 2 3 x . The 3 x is the product of x and 3, which are the terms that produce the perfect squares.

x 2 + 6 x + 9 = ( x + 3 ) 2

Got questions? Get instant answers now!

x 4 - 10 x 2 y 3 + 25 y 6 . This expression is a perfect square trinomial. The x 4 and 25 y 6 are both perfect squares. The terms that when squared produce x 4 and 25 y 6 are x 2 and 5 y 3 , respectively.

The middle term - 10 x 2 y 3 is divisible by 2 . In fact, - 10 x 2 y 3 2 = - 5 x 2 y 3 . Thus,

x 4 - 10 x 2 y 3 + 25 y 6 = ( x 2 - 5 y 3 ) 2

Got questions? Get instant answers now!

x 2 + 10 x + 16 . This expression is not a perfect square trinomial. Although the middle term is divisible by 2 , ? 10 x 2 = 5 x , the 5 and x are not the terms that when squared produce the first and last terms. (This expression would be a perfect square trinomial if the middle term were 8 x .)

Got questions? Get instant answers now!

4 a 4 + 32 a 2 b - 64 b 2 . This expression is not a perfect square trinomial since the last term - 64 b 2 is not a perfect square (since any quantity squared is always positive or zero and never negative).

Thus, 4 a 4 + 32 a 2 b - 64 b 2 cannot be factored using this method.

Got questions? Get instant answers now!

Practice set c

Factor, if possible, the following trinomials.

m 2 - 8 m + 16

( m - 4 ) 2

Got questions? Get instant answers now!

k 2 + 10 k + 25

( k + 5 ) 2

Got questions? Get instant answers now!

4 a 2 + 12 a + 9

( 2 a + 3 ) 2

Got questions? Get instant answers now!

9 x 2 - 24 x y + 16 y 2

( 3 x - 4 y ) 2

Got questions? Get instant answers now!

2 w 3 z + 16 w 2 z 2 + 32 w z 3

2 w z ( w + 4 z ) 2

Got questions? Get instant answers now!

x 2 + 12 x + 49

not possible

Got questions? Get instant answers now!

Exercises

For the following problems, factor the binomials.

a 2 9

( a + 3 ) ( a 3 )

Got questions? Get instant answers now!

x 2 16

( x + 4 ) ( x 4 )

Got questions? Get instant answers now!

a 2 100

( a + 10 ) ( a 10 )

Got questions? Get instant answers now!

4 a 2 64

4 ( a + 4 ) ( a 4 )

Got questions? Get instant answers now!

3 x 2 27

3 ( x + 3 ) ( x 3 )

Got questions? Get instant answers now!

4 a 2 25

( 2 a + 5 ) ( 2 a 5 )

Got questions? Get instant answers now!

36 y 2 25

( 6 y + 5 ) ( 6 y 5 )

Got questions? Get instant answers now!

12 a 2 75

3 ( 2 a + 5 ) ( 2 a 5 )

Got questions? Get instant answers now!

8 y 2 50

2 ( 2 y + 5 ) ( 2 y 5 )

Got questions? Get instant answers now!

x 2 y 2 25

( x y + 5 ) ( x y 5 )

Got questions? Get instant answers now!

x 4 y 4 9 a 2

( x 2 y 2 + 3 a ) ( x 2 y 2 3 a )

Got questions? Get instant answers now!

4 a 2 b 2 9 b 2

b 2 ( 2 a + 3 ) ( 2 a 3 )

Got questions? Get instant answers now!

a 2 b 2

( a + b ) ( a b )

Got questions? Get instant answers now!

x 4 y 4

( x 2 + y 2 ) ( x + y ) ( x y )

Got questions? Get instant answers now!

a 8 y 2

( a 4 + y ) ( a 4 y )

Got questions? Get instant answers now!

b 6 x 4

( b 3 + x 2 ) ( b 3 x 2 )

Got questions? Get instant answers now!

25 a 2

( 5 + a ) ( 5 a )

Got questions? Get instant answers now!

100 36 b 4

4 ( 5 + 3 b 2 ) ( 5 3 b 2 )

Got questions? Get instant answers now!

x 4 16

( x 2 + 4 ) ( x + 2 ) ( x 2 )

Got questions? Get instant answers now!

a 4 b 4

( a 2 + b 2 ) ( a + b ) ( a b )

Got questions? Get instant answers now!

x 12 y 12

( x 6 + x 6 ) ( x 3 + y 3 ) ( x 3 y 3 )

Got questions? Get instant answers now!

a 3 c 2 25 a c 2

a c 2 ( a + 5 ) ( a 5 )

Got questions? Get instant answers now!

a 4 b 4 c 2 d 2 36 x 2 y 2

Got questions? Get instant answers now!

49 x 2 y 4 z 6 64 a 4 b 2 c 8 d 10

( 7 x y 2 z 3 + 8 a 2 b c 4 d 5 ) ( 7 x y 2 z 3 8 a 2 b c 4 d 5 )

Got questions? Get instant answers now!

For the following problems, factor, if possible, the trinomials.

y 2 20 y 100

( y + 10 ) 2

Got questions? Get instant answers now!

x 2 10 x 25

( x 5 ) 2

Got questions? Get instant answers now!

a 2 24 a 144

( a 12 ) 2

Got questions? Get instant answers now!

9 x 2 6 x 1

( 3 x + 1 ) 2

Got questions? Get instant answers now!

16 a 2 24 a 9

( 4 a 3 ) 2

Got questions? Get instant answers now!

9 x 2 6 x y y 2

( 3 x + y ) 2

Got questions? Get instant answers now!

36 a 2 60 a b 25 b 2

( 6 a + 5 b ) 2

Got questions? Get instant answers now!

12 a 2 60 a 75

3 ( 2 a 5 ) 2

Got questions? Get instant answers now!

32 x 2 16 x 2

2 ( 4 x + 1 ) 2

Got questions? Get instant answers now!

4 a 2 a 9

not factorable

Got questions? Get instant answers now!

x 5 8 x 4 16 x 3

x 3 ( x + 4 ) 2

Got questions? Get instant answers now!

12 a 3 b - 48 a 2 b 2 + 48 a b 3

Got questions? Get instant answers now!

Exercises for review

( [link] ) Factor ( m 3 ) x ( m 3 ) y .

( m 3 ) ( x y )

Got questions? Get instant answers now!

( [link] ) Factor 8 x m 16 x n 3 y m 6 y n by grouping.

Got questions? Get instant answers now!

Questions & Answers

it is the relatively stable flow of income
Chidubem Reply
what is circular flow of income
Divine Reply
branches of macroeconomics
SHEDRACK Reply
what is Flexible exchang rate?
poudel Reply
is gdp a reliable measurement of wealth
Atega Reply
introduction to econometrics
Husseini Reply
Hi
mostafa
hi
LEMLEM
hello
Sammol
hi
Mahesh
bi
Ruqayat
hi
Ruqayat
Hi fellas
Nyawa
hey
Sammol
hi
God
hello
Jahara
Good morning
Jorge
hi
abubakar
hi
Nmesoma
hi
Mahesh
Hi
Tom
Why is unemployment rate never zero at full employment?
Priyanka Reply
bcoz of existence of frictional unemployment in our economy.
Umashankar
what is flexible exchang rate?
poudel
due to existence of the pple with disabilities
Abdulraufu
the demand of a good rises, causing the demand for another good to fall
Rushawn Reply
is it possible to leave every good at the same level
Joseph
I don't think so. because check it, if the demand for chicken increases, people will no longer consume fish like they used to causing a fall in the demand for fish
Anuolu
is not really possible to let the value of a goods to be same at the same time.....
Salome
Suppose the inflation rate is 6%, does it mean that all the goods you purchase will cost 6% more than previous year? Provide with reasoning.
Geetha Reply
Not necessarily. To measure the inflation rate economists normally use an averaged price index of a basket of certain goods. So if you purchase goods included in the basket, you will notice that you pay 6% more, otherwise not necessarily.
Waeth
discus major problems of macroeconomics
Alii Reply
what is the problem of macroeconomics
Yoal
Economic growth Stable prices and low unemployment
Ephraim
explain inflationcause and itis degre
Miresa Reply
what is inflation
Getu
increase in general price levels
WEETO
Good day How do I calculate this question: C= 100+5yd G= 2000 T= 2000 I(planned)=200. Suppose the actual output is 3000. What is the level of planned expenditures at this level of output?
Chisomo Reply
how to calculate actual output?
Chisomo
how to calculate the equilibrium income
Beshir
Criteria for determining money supply
Thapase Reply
who we can define macroeconomics in one line
Muhammad
Aggregate demand
Mohammed
C=k100 +9y and i=k50.calculate the equilibrium level of output
Mercy Reply
Hi
Isiaka
Hi
Geli
hy
Man
👋
Bahunda
hy how are you?
Man
ys
Amisha
how are you guys
Sekou
f9 guys
Amisha
how are you guys
Sekou
ys am also fine
Amisha
fine and you guys
Geli
from Nepal
Amisha
nawalparasi district from belatari
Amisha
nd u
Amisha
I am Camara from Guinea west Africa... happy to meet you guys here
Sekou
ma management ho
Amisha
ahile becheclor ho
Amisha
hjr ktm bta ho ani k kaam grnu hunxa tw
Amisha
belatari
Amisha
1st year ho
Amisha
nd u
Amisha
ahh
Amisha
kaha biratnagar
Amisha
ys
Amisha
kina k vo
Amisha
money as unit of account means what?
Kalombe
A unit of account is something that can be used to value goods and services and make calculations
Jim
all of you please speak in English I can't understand you're language
Muhammad
I want to know how can we define macroeconomics in one line
Muhammad
it must be .9 or 0.9 no Mpc is greater than 1 Y=100+.9Y+50 Y-.9Y=150 0.1Y/0.1=150/0.1 Y=1500
Kalombe
Mercy is it clear?😋
Kalombe
hi can someone help me on this question If a negative shocks shifts the IS curve to the left, what type of policy do you suggest so as to stabilize the level of output? discuss your answer using appropriate graph.
Galge Reply
if interest rate is increased this will will reduce the level of income shifting the curve to the left ◀️
Kalombe
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Elementary algebra. OpenStax CNX. May 08, 2009 Download for free at http://cnx.org/content/col10614/1.3
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Elementary algebra' conversation and receive update notifications?

Ask